Question

How do you find the greatest common factor of 40, 10?

Answer 1

Hint: We will first do the prime factorization of each of the given numbers. Then, we will see how many numbers come common in them and then multiply the common numbers to get the greatest common factor.

Complete step-by-step answer:
We have the numbers 40 and 10 whose greatest common divisor we are required to find.
Now, we will write the prime factorization of 40.
$ \Rightarrow 40 = 2 \times 2 \times 2 \times 5$ …………….(1)
Now, we will write the prime factorization of 10.
$ \Rightarrow 10 = 2 \times 5$ …………….(2)
In the equations (1) and (2), if we notice the right hand side carefully, we see that they have 2 and 5 in common. Basically, the number 40 has 2 2’s extra than the same prime factorization of 10 comes in the picture. In 40, we have three 2’s and one 5 and in 10, we have one 2 and one 5.
Hence, the greatest common factor of 40 and 10 is $2 \times 5$.

Therefore, the greatest common factor of 40 and 10 is 10.

Note:
The students must note that there are alternate ways to do the same. Let us see them to know what ways we can use to do the same thing as we did above.
Alternate way 1:
Since, we know that we can write 40 as a multiple of 10 and 4. Now, we can write that as a multiple of 10 and 1. We can see that they have 10 as a common multiple. Hence, 10 is the required greatest common factor.
Alternate way 2:
Now, we can also do this as: when we divide 40 by 10, we get the quotient of 4 and remainder of 0.
So, if we use Euclid’s division algorithm to see the same, we get the largest divisor as the greatest common divisor and thus we have the greatest common divisor as 10.